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How Set Theory and Quantum How Set Theory and Quantum Physics Can Give Us a Scientific Physics Can Give Us a Scientific

Concept of ConsciousnessConcept of Consciousness

J. Andrew RossToward a Science of ConsciousnessApril 8–12, 2002, Tucson, Arizona

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Conscious brains create knowledge

Human consciousness is created by brain activityConscious states are correlated with brain statesConscious human beings generate knowledge

The brain

The seat ofsubjectivity

The body

Transition toobjectivity

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Propositions express knowledge

Conscious states are states of knowledgeEpistemology is the theory of knowledgeOntology is the theory of what existsKnowledge states are propositional

Bivalent

Truths Propositions Falsehoods

P ∨ ¬ P

¿ "This proposition is false" ?

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Knowledge states form trees

As time passes and knowledge developsMeaning and truth conditions changeDecision and proof procedures changeThe tree of knowledge grows

Root node

Leaf nodes

Decision nodes

Unfolding of meaningand truth conditions

Development ofdecision and

proof procedures

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Conscious states are logic states

A logical language can be any symbolic medium used by a conscious subjectA model for the language can be any world that surrounds the subject

Medium Model

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Are brains computers?

ComputersHave digitized input and outputHave a finite number of inner statesOperate according to fixed rulesAre classical machines

Human brainsHave approximately digitized input/output Have a vast but probably finite number of inner states Operate according to rulesthat are presumably fixed Are subject to quantum physics

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Brains are natural neuronets

The human cerebral cortex contains some hundred billion neuronsAn average neuron connects with thousands of other neuronsNeurons receive and emit electrical signals

+50

–100

mV 0

1 ms

A neural signal

Receivesignalsfrommany

Emitsignals

to afew

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ANNs may facea fundamentalphysical barrier

Artificial neuronets are computers

Artificial neural networks can solve logic problemsThey can learn by trial and error They can emulate many brain functions

But can ANNs emulate brains completely?

=?Classicalmachine Quantum?

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Subjects comprehend objects

Sets are elements from above, classes from below

Elements stand for objectsClasses stand for subjects

Elements xObjects

Class CSubject

Mindsets!

Can we see a setas a formal

metaphor for amoment in theongoing life ofconsciousness?

x ∈ C

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Comprehension creates mindsets

Zermelo–Fraenkel set theoryZF axioms: For all x, y ∈ V,

Extensionality: x = y ↔ (∀z)(z ∈ x ↔ z ∈ y)Regularity: x ≠ ∅ → (∃z)(z ∈ x ∧ z ∩ x = ∅)Pairs: {x, y} ∈ VUnion: If U(x) = {u | (∃v)(u ∈ v ∧ v ∈ x)} then U(x) ∈ VPower set: If P(x) = {u | u ⊆ x} then P(x) ∈ VNull set: ∅ ∈ VInfinity: If ω = {u | ∅ ∈ u ∧ (∀v)(v ∈ u → v ∪ {v} ∈ u)} then ω ∈ VReplacement schema:For any ZF function f from D to C, D ∈ V → C ∈ V

Zermelo–Fraenkel set theoryZF axioms: For all x, y ∈ V,

Extensionality: x = y ↔ (∀z)(z ∈ x ↔ z ∈ y)Regularity: x ≠ ∅ → (∃z)(z ∈ x ∧ z ∩ x = ∅)Pairs: {x, y} ∈ VUnion: If U(x) = {u | (∃v)(u ∈ v ∧ v ∈ x)} then U(x) ∈ VPower set: If P(x) = {u | u ⊆ x} then P(x) ∈ VNull set: ∅ ∈ VInfinity: If ω = {u | ∅ ∈ u ∧ (∀v)(v ∈ u → v ∪ {v} ∈ u)} then ω ∈ VReplacement schema:For any ZF function f from D to C, D ∈ V → C ∈ V

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Sets form a cumulative hierarchy

Every ZF set x has an ordinal rank R(x)Ordinal numbers α – John von Neumann

0 = ∅ = { }α = {β | β < α}

V-sets Vα

V0 = 0Vα = P(Vα −1) for successor ordinals αVλ = U {Vα | α < λ} for limit ordinals λ

R(x) = the least ordinal αsuch that x ⊆ Vα

Ranks of V-sets form cascades of infinities

ZF

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The universe of sets is transfinite

Ordinalrank

α

Absoluteinfinity

ω

V

Vω

Vα

Hereditarilyfinite sets

Transfinitesets

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Ranks of sets accumulate logically

Ranks in V form models for first order theories

a1 a2 a3

Elementsof rank αf1 f2

C1 = {x | f1(x)} = {a1, a2, a3}

C2 = {x | f2(x)} = {a2}

Classes ofrank α + 1

Ontologyof elements

Epistemologyof classes

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© 2002 J.A.Ross 14Development of a consciousness

The universe of knowledge evolves

Epistemology and ontology form a dialectic in V

Ontology αEpistemology α − 1

Ontology α + 1Epistemology α

Progress

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Knowledge states form worlds

A knowledge state isA totality of facts – Ludwig Wittgenstein

A set of true propositions Tractatus

Closed under logical inferenceSatisfied in a world

New facts are informative

Worldafter

Knowledgestate β

Advancefrom α to β

Knowledgestate α

Worldbefore

New facts

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Worlds are like virtual realities

A world embeds a subjectThe world is reality for the embedded subject

A world may be actual or possibleAn actual world is an existing state of

Information (bits)Knowledge (facts)Consciousness (qualia)

A possible world is a virtual reality

The VR is defined by computation fromatomic bits– David Deutsch

VR

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Worlds can be actual or possible

Modal logic is the logic of possible worldsThe actual world G is the world as it is nowPossible worlds W are worlds as they may beAn accessibility relation R links pairs of worlds

– Saul Kripke

W

WW

W WR

R RR

R G

Necessarily P is true in G iff, for all worlds W accessible from G, P is true in W

Possibly P is true in G iff, for some world W

accessible from G, P is true in W

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Modalities can be epistemic or ontic

Axioms for modal logic define Necessarily P: PPossibly P: P

In a modal theory, modalities may be

EpistemicP if P is implied by what is knownP if P is consistent with what is known

OnticP if the intrinsic probability of P = 1P if the intrinsic probability of P > 0

Psychological

Physical

Fuzzy distinction© 2002 J.A.Ross 18

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Probability can be epistemic or ontic

In classical physics, the world is eternalReality evolves rigidly along a fixed timeline Exact laws determine the past and future Statistical approximations generate probabilities

Classical probabilities are epistemic

In quantum physics, the world is changingReality comes into focus along a growing timelineThe past is fixed but the future is fuzzyThe probability of possible futures is intrinsic

Quantum probabilities are ontic

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Classical states form a continuum

In classical physics, a state of a system S is a definite configuration of the parts of S

Gas molecules ina closed volume

Each moleculehas a definitemass, position,velocity, ...

DETERMINISMIn principle,

given state S1 at time t1,state S2 at any later time t2

can be predicted

CHAOSIn fact,

any errors in measuring S1grow so fast that soon S2

cannot be predicted

Weatherforecasting– Edward Lorenz

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Quantum states are discrete

Quantization generates uncertaintyPlanck's quantum of action h(about 6 • 10–34 joule-second) is a tiny fuzzball of uncertainty

In quantum theory, particles can appear or disappear randomly

In trying to predict the behavior of a systemof particles, the best we can do is calculate the probabilities of creation or annihilationat each point in spacetime

∆p or ∆E∆x or ∆t

∆p ∆x ~ h∆E ∆t ~ h

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Wave-particleduality implies

uncertainty– Werner

Heisenberg

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Possible states define spaces

A world is a state of a physical systemAn actual world G is a real state of a systemA possible world W is a virtual state of a system

Each observable state of a physical system forms a dimension in a mathematical state space

State vectorspecifies thestate of the

system by itsdirection

(observablestates are

orthogonal)

State spacerepresents

all observablestates of the

system asdimensions

(number maybe infinite)

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Quantum states can be superposed

A system can be in several states at onceGenerally, it is in a superposition or mixed state of the possible observed values for an observable QEach dimension of the state space is a pure state of Q

Measurement, observation, or interaction nudges a mixed state to a pure state

Mixed state instate space

Pure state instate space

MeasurementInteraction

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Quantum systems decohere stepwise in time during interaction with their environment

For objects of mass > 1 fg (mass of a small grain of dust) decoherence times < 1 as (time for light to cross an atom)

Quantum superpositions decohere

Superposition of statesFor each state,

old probability < 1

Measured stateFor this state,

new probability = 1

Old world: time t New world: t + ∆t

Measurement

Interaction

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Physical worlds unfold in time

Quantum systems evolve in timeSuperpositions decohere stepwise to pure statesMoments of time are realized by approximately simultaneous devirtualization of fuzzy quanta

Momentof time

Realizationof quanta

Simultaneityis fuzzy∆t > 0

Quanta varyin duration∆E ∆t ~ h

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Both time and space unfold

Space and time are inseparable – Albert Einstein

If time unfolds, space does too

x = ct

Timelikeintervals

Spacelikeintervals

Futurelight conesmall and

soft

Pastlight conelarge and

hard

Time t

Space x, y, z

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Time can be ontic or epistemic

Physicaluniverse

Ontic timeIs defined asclock time inbasic physicsIs our bestconceptionof real time

Epistemic timeIs experiencedas a flux of nowstatesIs real only now

Oursocialworld

MyworldNow

Realtime

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What you see is what you use …

PhenomenologyWYSIWY use to build a theory of realityThe thinker thinks in a self-collapsing worldInner access is no more privileged than outer accessThe thinker is an artifact of "his" own phenomenology

The thinker iscocrystalized withthe landscape

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… therefore I am conscious

The thinker creates an evolving VR(to help survive in a natural world)ThereforeI am conscious

CogitoErgoSum

– René Descartes

Our

world Yourworld

Myworld

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The conscious brain – David Chalmers

From inside, seems like a phenomenal world of qualia

From outside, seems like a wet pulsating lump

These views are worlds apart

How can I see my brain?

InsideFirst-person

outlook

OutsideThird-person

insight

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I'm living in a loop

The inner I looks out bloop

And looking back sees me floop

All in all, quite strange gloop

– DouglasHofstadter

To infinity …First-person

outlook

… and backThird-person

insight

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Brains realize quantum states

Biological processes occur at molecular scalesAt molecular scales quantum effects can dominateNeuronets learn by thermodynamic relaxation Relaxation is a stochastic process In the brain, it is an extremelydelicate analog process

Brain states may show quantum effects

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Conscious states have rhythm

Conscious states evolve in moments of nowLarge patches of phenomenal reality decoherewith a periodicity that seems more or less steadyConscious states are phenomenal equivalence classesof brain states experienced from the insideAn increment of now ∆t ~ 20 – 100 ms in a band of frequencies in the decahertz range around

The flicker fusion rateA fast reaction timePhysiological tremor

0 0 0 0 00 11 011 0 0101 11 0 0 011 011 0 011 011 0 0 0 0f (now) ~ 12 Hzf (now) ~ 12 Hz

Timeness isconsciousness

– RodolfoLlinás

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Conscious states are unified

Consciousness is unified – how, physically?Like a laser beam?

Photons lose their identitiesin a boson condensate

Eachstate isunified

A boson condensate is a Bose–Einstein (BE) state where the separate identities of the constituent particles are dissolved in a quantum unityThis is the only known way tophysically unify brain events

– Scott Hagan34

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Do biophotons unify life processes?

Cells in the body exchange photons – Fritz Popp

These photons Are mostly microwave or infrared and sometimes visible lightMay communicate biologically useful information

? Is it possible thatTransient coherent states of these photons coordinate and unify life processes?A hierarchy of such states leads seamlessly to photonic states supporting consciousness?

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Brainwaves correlate with consciousness

Consciousness is correlated with extended decahertz electromagnetic (EM) brainwavesSynchronized neural firings create coherent EM fields in regions ~ 1 ml with frequencies ~ 40 HzThese gamma waves generate neural bindingand unified percepts in consciousness

– Wolf Singer

CoherentdecahertzEM fields

Expandingenvelope

wavefronts FACT

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Thalamocortical loops mark time

Consciousness is correlated with temporal binding of neural groups firing in decahertz rhythmsThalamocortical loops firing rhythmically form a main mechanism of brain functionThese loops unify isochronous conscious states

– Rodolfo Llinás

Cortex

Thalamus

Thalamo-corticalloops FACT

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Is consciousness photonic?

Interneural photons with f ~ 40 Hz that form boson condensates lasting for 1 now are the quantum correlates of consciousness

– Andrew Ross

Unstable BE statesof photonsserve asmomentarymirrors for ourstates ofmind

Our statesof mind

are frozenin photons

Timestands

still for aphoton

– Albert Einstein

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Do brainwaves form a quantum foam?

Synchronous neural firings emit waves of photonsThe photons form bubbles of superposed states that extend for ~ 80 ms over thethalamocortical systemAs a bubble pops, it

Freezes a moment of nowReflects qualia like a mirrorRealizes a state of mind

Popping bubbles form a quantum foam

Foaming decahertz photons have uncertainties ∆t ~ 30 ms

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Mindworlds 'r' us

Mindworlds are structured sets of qualia with subjective sides that are

Phenomenologically closed and unifiedManifested as consistent sets of factsTemporally transient or momentaryExperienced as states of an ongoing I

The corresponding objective sides areCentered on living and functioning brainsAssociated with specific interneural activityRealized in a foam of photon bubblesLinked in the flow of an ongoing me

?

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